Atomic Spectral Structure from Time-Scalar Field Theory

We derive the structural form of atomic bound-state spectra from Time-Scalar Field Theory (TSFT) without postulating quantum mechanics or a Coulomb potential. Starting from the scalar-time field equation, we construct localized, static background solutions with asymptotic behavior Θ₀ (r) = Θ_∞ + A/r. Expanding the induced operator V ′′ (Θ₀ (r) ) about Θ_∞ yields an emergent inverse-radial interaction −κ/r at large distances, where κ is fixed by the background amplitude and the third derivative of the scalar-time potential. The absolute normalization of κ is obtained from the recovered electrodynamic sector of TSFT. Using E_Θ = −∇Θ together with the inhomogeneous Maxwell equation ∇·E_Θ = ρq/ε₀, Gauss-law normalization determines the far-field coefficient A in terms of the conserved source charge. This yields κ = CZα, with C fixed by the scalartime potential curvature and electrodynamic normalization, and α inherited from the closure structure of the theory. The resulting radial problem is a Sturm–Liouville system with inverse-radial leading behavior. Imposing normalizability yields Laguerre termination and a discrete spectrum εₙ = −κ²/ (4n²) with shell capacities 2n². Subleading 1/r² corrections generate subshell-dependent shifts Δₙℓ treated perturbatively. Thus, the principal organization of atomic spectra arises from scalar-time field dynamics together with the internally recovered electrodynamic normalization, providing a non-circular route to the hydrogenic spectral structure.